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Damanvir Singh Binner

Publications and source records attributed to Damanvir Singh Binner.

13 recordsLinked to original sources
arXiv2022

Berkovich-Uncu type Partition Inequalities Concerning Impermissible Sets and Perfect Power Frequencies

Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697-728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263-284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect t^{th} power. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case t=1. To prove these inequalities, our methods involve constructing injective maps between the relevant sets of partitions. The construction of these maps crucially involves concepts from analysis and calculus, such as explicit maps used to prove countability of N^t, and Jensen's inequality for convex functions, and then merge them with techniques from number theory such as Frobenius numbers, congruence classes, binary numbers and quadratic residues. We also show a connection of our results to colored partitions. Finally, we pose an open problem which seems to be related to power residues and the almost universality of diagonal ternary quadratic forms.

math.CO↗
arXiv2022

On $k$-measures and Durfee squares of partitions

Recently, Andrews, Bhattacharjee and Dastidar introduced the concept of $k$-measure of an integer partition, and proved a surprising identity that the number of partitions of $n$ which have $2$-measure $m$ is equal to the number of partitions of $n$ with a Durfee square of side $m$. The authors asked for a bijective proof of this result and also suggested a further exploration of the properties of the number of partitions of $n$ which have $k$-measure $m$ for $k \geq 3$. In this note, we complete these tasks. That is, we obtain a short combinatorial proof of the result of Andrews, Bhattacharjee and Dastidar, and using this proof, we easily generalize this result for $k$-measures.

math.CO↗
arXiv2022

A Comparison of Integer Partitions Based on Smallest Part

For positive integers $n, L$ and $s$, consider the following two sets that both contain partitions of $n$ with the difference between the largest and smallest parts bounded by $L$: the first set contains partitions with smallest part $s$, while the second set contains partitions with smallest part at least $s+1$. Let $G_{L,s}(q)$ be the generating series whose coefficient of $q^n$ is difference between the sizes of the above two sets of partitions. This generating series was introduced by Berkovich and Uncu in 2019. Previous results concentrated on the nonnegativity of $G_{L,s}(q)$ in the cases $s=1$ and $s=2$. In the present paper, we show the eventual positivity of $G_{L,s}(q)$ for general s and also find a precise nonnegativity result for the case $s=3$.

math.CO↗
arXiv2022

Number of partitions of n with a given parity of the smallest part

We obtain a combinatorial proof of a surprising weighted partition equality of Berkovich and Uncu. Our proof naturally leads to a formula for the number of partitions with a given parity of the smallest part, in terms of S(i), the number of partitions of i into distinct parts with even rank minus the number with odd rank, for which there is an almost closed formula by Andrews, Dyson and Hickerson. This method of calculating the number of partitions of n with a given parity of the smallest part is practical and efficient.

math.CO↗
arXiv2021

Proofs of Chappelon and Alfons\'ın Conjectures On Square Frobenius Numbers and its Relationship to Simultaneous Pell's Equations

Recently, Chappelon and Alfons\'ın defined the square Frobenius number of coprime numbers $m$ and $n$ to be the largest perfect square that cannot be expressed in the form $mx+ny$ for nonnegative integers $x$ and $y$. When $m$ and $n$ differ by $1$ or $2$, they found simple expressions if neither $m$ nor $n$ is a perfect square. If either $m$ or $n$ is a perfect square, they formulated some interesting conjectures which have an unexpected close connection with a known recursive sequence, related to the denominators of Farey fraction approximations to $\sqrt{2}$. In this note, we prove these conjectures. Our methods involve solving Pell's equations $x^2-2y^2=1$ and $x^2-2y^2=-1$. Finally, to complete our proofs of these conjectures, we eliminate several cases using a bunch of results related to solutions of simultaneous Pell's equations.

math.NT↗
arXiv2021

A Complete List of all Numbers not of the Form $ax+by$

For given coprime positive integers $a$ and $b$, the classical Frobenius coin problem asked to find the largest number that cannot be expressed in the form $ax+by$ for nonnegative integers $x$ and $y$, also known as the Frobenius number. Sylvester answered this question and also discovered the number of these nonrepresentable numbers. In recent times, a lot of progress has been made regarding the sums of a fixed power of these nonrepresentable numbers, commonly known as the Sylvester sums. However, the actual list of nonrepresentable numbers has remained mysterious so far. In this note, we obtain a complete list, that is a simple explicit description of these nonrepresentable numbers. We give two different proofs of our result. The first one enables us to use Eisenstein's lattice point counting techniques to study the set of nonrepresentable numbers, whereas the second one is a short direct proof.

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arXiv2021

On Conjectures Concerning the Smallest Part and Missing Parts of Integer Partitions

For positive integers $s$ and $L \geq 3$, Berkovich and Uncu (Ann. Comb. $23$ ($2019$) $263$--$284$) conjectured an inequality between the sizes of two closely related sets of partitions whose parts lie in the interval $\{s, \ldots, L+s\}$. Further restrictions are placed on the sets by specifying impermissible parts as well as a minimum part. The authors proved their conjecture for the cases $s=1$ and $s=2$. In the present article, we prove the conjecture for general $s$ by proving a stronger theorem. We also prove other related conjectures found in the same paper.

math.CO↗
arXiv2021

Generalization of a Result of Sylvester Regarding the Frobenius Coin Problem and an Elementary Proof of Eisenstein's Lemma for Jacobi Symbols

In a recent work, the present author generalized a fundamental result of Gauss related to quadratic reciprocity, and also showed that the above result of Gauss is equivalent to a special case of a well-known result of Sylvester related to the Frobenius coin problem. In this note, we use this equivalence to show that the above generalization of the result of Gauss naturally leads to an interesting generalization of the result of Sylvester. To be precise, for given positive coprime integers $a$ and $b$, and for a family of values of $k$ in the interval $0 \leq k < (a-1)(b-1)$, we find the number of nonnegative integers $\leq k$ that can be expressed in the form $ax+by$ for nonnegative integers $x$ and $y$. We also give an elementary proof of Eisenstein's Lemma for Jacobi symbols using floor function sums. Our proof provides a natural straightforward generalization of the Gauss-Eisenstein proof of the law of quadratic reciprocity for Jacobi symbols.

math.NT↗
arXiv2021

Some Bounds for Number of Solutions to $ax + by + cz = n$ and their Applications

In a recent work, the present author developed an efficient method to find the number of solutions of $ax+by+cz=n$ in non-negative integer triples $(x,y,z)$ where $a,b,c$ and $n$ are given natural numbers. In this note, we use that formula to obtain some simple looking bounds for the number of solutions of $ax+by+cz=n$. Using these bounds, we solve some special cases of a problem related to the generalization of Frobenius coin problem in three variables. Moreover, we use these bounds to disprove a recent conjecture of He, Shiue and Venkat regarding the solution structure of $ax+by+cz=n$.

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arXiv2019

Number of Non-negative Integer Solutions of the Equation ax+by+cz=n and its Relation with Quadratic Residues

In 2000, A. Tripathi used generating functions to obtain a formula for the number of non-negative solutions (x,y) of the equation ax + by = n where a, b and n are given positive integers. We generalize this procedure for the number of solutions of the equation ax + by + cz = n. The formula leads us to a surprising connection between the number of solutions of the equation ax + by + cz = n and quadratic residues. As a consequence of our work, we are able to prove the equivalence between two fundamental results by Gauss and Sylvester in the nineteenth century which are generally viewed to be independent.

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